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Thread: Not completely understanding what analytic means. HELP!

  1. #1
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    Exclamation Not completely understanding what analytic means. HELP!

    I understand the following:
    Theorem: Suppose the real functions $\displaystyle u(x,y)$ and $\displaystyle v(x,y)$ are continuous and have continuous 1st order partials in a domain D. If $\displaystyle u$ and $\displaystyle v$ satisfy the CR equations at all points in D, then $\displaystyle f(z) = u(x,y) +iv(x,y)$ is analytic in D.

    Therefore, the CR equations $\displaystyle u_{x} = v_{y}$ and $\displaystyle u_{y}=-v_{x}$ must hold.

    But I don't understand what I'm supposed to conclude when I do the following problem:
    $\displaystyle f(z)=|z-10|^2 = y^2 + (x^2 - 10)^2$

    When I do the CR equations I get $\displaystyle v_{y} = 0$ and $\displaystyle v_x=0$. Does that mean it is only differentiable at $\displaystyle (0,0)$? What does that say about being analytic? I've seen a few problems like this were some of the CR equations are equal to zero such as $\displaystyle f(z) = |z|^2$. For that one I understand that limits at different paths are not equal therefore only differentiable at $\displaystyle z = 0$. Do I need to do different paths to show where it's only differentiable for the problem above or how should I try to tackle this problem? Thank you so much!
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  2. #2
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    Re: Not completely understanding what analytic means. HELP!

    "Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at $\displaystyle z= z_0$, if and only if it is equal to its Taylor's polynomial in some open neighborhood of $\displaystyle z_0$, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
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  3. #3
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    Re: Not completely understanding what analytic means. HELP!

    Quote Originally Posted by HallsofIvy View Post
    "Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at $\displaystyle z= z_0$, if and only if it is equal to its Taylor's polynomial in some open neighborhood of $\displaystyle z_0$, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
    Okay, I've read my notes a few more times and I think I'm beginning to understand. For the problem I stated I got $\displaystyle u_x = 2x - 20$, $\displaystyle u_y = 2y$, $\displaystyle v_x = 0$, and finally $\displaystyle v_y = 0$. Therefore, $\displaystyle 2x -20 = 0$ and $\displaystyle 2y = 0$. Therefore, it is only differentiable at $\displaystyle (0,0)$. So how do I jump to the conclusion that it is (or it isn't) analytic?
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    Re: Not completely understanding what analytic means. HELP!

    Quote Originally Posted by HallsofIvy View Post
    "Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at $\displaystyle z= z_0$, if and only if it is equal to its Taylor's polynomial in some open neighborhood of $\displaystyle z_0$, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
    And by paths I meant taking the limit along the real axis or the imaginary axis or any other point. If both limits aren't the same then the limit of f(z) does not exist.
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  5. #5
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    Re: Not completely understanding what analytic means. HELP!

    I've read my notes over and over again and I'm still stuck. Can anyone help me out?
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